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Interval Finite Element Method with MATLAB.pdf

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Front-Matter_2018_Interval-Finite-Element-Method-with-MATLAB.pdf (p.1-2)
Title page
Copyright_2018_Interval-Finite-Element-Method-with-MATLAB.pdf (p.3)
Copyright
Author-Biographies_2018_Interval-Finite-Element-Method-with-MATLAB.pdf (p.4)
Preface_2018_Interval-Finite-Element-Method-with-MATLAB.pdf (p.5-7)
Acknowledgments_2018_Interval-Finite-Element-Method-with-MATLAB.pdf (p.8)
nayak2018.pdf (p.9-14)
Interval Arithematic
nayak2018 (1).pdf (p.15-25)
Interval Finite Element Method
nayak2018 (2).pdf (p.26-40)
Preliminaries of MATLAB
nayak2018 (3).pdf (p.41-51)
One-Dimensional Interval Finite Element
nayak2018 (4).pdf (p.52-66)
MATLAB® Code for One-Dimensional Interval Finite Element
nayak2018 (5).pdf (p.67-82)
Two-Dimensional Interval Finite Element
nayak2018 (6).pdf (p.83-108)
MATLAB® Code for Two-Dimensional Interval Finite Element
nayak2018 (7).pdf (p.109-122)
Three-Dimensional Interval Finite Element
nayak2018 (8).pdf (p.123-154)
10.1016@B978-0-12-812973-9.09992-6.pdf (p.155-158)
Index
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INTERVAL FINITE ELEMENT METHOD WITH MATLAB®
INTERVAL FINITE ELEMENT METHOD WITH MATLAB® SUKANTA NAYAK SNEHASHISH CHAKRAVERTY
Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2018 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-12-812973-9 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisition Editor: Brian Guerin Editorial Project Manager: Katie Chan Production Project Manager: Anusha Sambamoorthy Cover Designer: Greg Harris Typeset by SPi Global, India
AUTHOR BIOGRAPHIES Dr. Sukanta Nayak is an assistant professor of mathematics at Amrita Vishwa Vidyapeetham, Coimbatore. Prior to coming to Amrita Vishwa Vidyapeetham, he was a postdoctoral research fellow at University of Johan- nesburg, South Africa, from August 2016 to June 2017. He was a lecturer at Veer Surendra Sai University of Technology, Burla, from January 2016 to May 2016. He received Ph.D. in mathematics from National Institute of Technology Rourkela in January 2016. He has received Global Excellence Stature Postdoctoral Fellowship in 2016 and postgraduate scholarship from Government of Odisha in 2008. In addition, he has qualified the all-India Graduate Aptitude Test in Engineering (GATE) and was awarded “Best Presentation” of Department of Mathematics at Research Scholar Week (RSW-2015) NIT Rourkela in 2015. His research interests span both uncertainty modeling and its engineering applications. He is also an author of the book entitled Neutron Diffusion: Concepts and Uncertainty Analysis for Engineers and Scientists of CRC Press (Taylor and Francis) and, until now, he has published 11 peer-reviewed research articles and 2 book chapters of international repute. Dr. Snehashish Chakraverty is a professor of mathematics in National Institute of Technology Rourkela, Odisha. He received Ph.D. from IIT Roorkee in 1992. Then he did postdoctoral research at ISVR, University of Southampton, United Kingdom, and at Concordia University, Canada. He was a visiting professor at Concordia & McGill Universities, Canada, and University of Johannesburg, South Africa also. Prof. Chakraverty has authored 10 books and published around 275 research papers in journals and conferences. He was President of the Section of Mathematical Sciences (including Statistics) of Indian Science Congress (2015–16) and was the Vice President—Orissa Mathematical Society (2011–13). Prof. Chakraverty is recipient of a few prestigious awards, viz. INSA International Bilateral Exchange Program, Platinum Jubilee ISCA Lecture, CSIR Young Scientist, BOYSCAST, UCOST Young Scientist, Golden Jubilee CBRI Director’s Award, Roorkee University gold Medals, etc. He has undertaken around 17 research projects as Principal Investigator funded by different agencies. His present research area includes soft computing, numerical analysis, differ- ential equations, mathematical modeling, vibration and inverse vibration problems. vii
PREFACE Finite element method is a well-known and versatile numerical method that is used to model and investigate various problems in different subject areas. On the other hand, uncertainty plays an important role in real-world problems. These uncertainties occurs due to the vague, imprecise, and incomplete information about the variables and parameters being a result of errors in measurement, observations, experiment, applying different operating conditions, or it may be maintenance-induced errors, which are uncertain in nature. To manage these uncertainties and vagueness, one may use interval or stochastic environment in parameters and variables in place of crisp (fixed) ones. However, probability theory requires sufficient number of data, which are not possible in general. Therefore, instead of sto- chastic, one may use intervals for the uncertain parameters and accordingly the problem may be modeled. Hence, the governing differential equations become interval. In order to investigate various uncertain practical prob- lems, the concept of interval uncertainties may be hybridized with the finite element method (FEM) to develop interval finite element method (IFEM). As such, this book aims to provide a new direction for the reader to deal with the uncertain problems by using IFEM. Various structural problems, viz. spring mass, bar, truss, frame, etc., have been modeled here using IFEM in uncertain (interval) environment. Further, a systematic approach with ® respect to MATLAB codes has been provided to study the mentioned problems using IFEM. The major essence of this book is to focus on uncertainties involved in various simple structural problems along with their numerical solutions and ® MATLAB codes. This book includes a systematic procedure, and chapters have been arranged in a sequence so that readers will follow it easily. Accord- ingly, the first three chapters consist of basics of interval arithmetic, hybrid- ® ization with FEM (i.e., IFEM) and that of MATLAB preliminaries. Next even-numbered chapters, viz. 4, 6, and 8, address the formulation of IFEM for various easy-to-follow structural problems and the odd-numbered chap- ters, viz. 5, 7, and 9, include the corresponding MATLAB codes. ® Let us elaborate few more details regarding the chapters now. As such, Chapter 1 describes various terminologies of interval uncertainties and inter- val arithmetic. In this chapter, a brief motivation of interval uncertainties has been introduced to reflect a concise idea of intervals. Chapter 2 has been ix
x Preface ® dedicated to understand IFEM for solving uncertain differential equations. The involved parameters in the governing differential equations have been considered as intervals and then general procedure is addressed to deduce ® IFEM. Preliminaries of MATLAB software, various syntax, commands, and operators are included in Chapter 3. It also includes a systematic ® approach to write the MATLAB codes. Chapter 4 deals with one- dimensional structural element formulations using IFEM. Considering interval uncertainties, stiffness matrices and elemental equations of various structural elements, viz. spring, bar, and quadratic bar elements, are devel- oped. Chapter 5 presents IFEM MATLAB functions for spring, bar, and quadratic bar elements, and various example problems have been investi- gated through the developed codes. Next, formulation of IFEM for two-dimensional structural elements, viz. plane truss, beam, plane frame, and triangular elements, is given in Chapter 6. Taking interval uncertainties, interval stiffness matrices and elemental equations are again formed for the ® same. In Chapter 7, the MATLAB codes for two-dimensional structural elements as mentioned in Chapter 6 are developed. The developed codes have been utilized to solve few example problems and corresponding results are reported. Further, the formulations of IFEM for three-dimensional ele- ments, viz. space truss, space frame, and linear tetrahedral elements, are described in Chapter 8. Then, the interval element stiffness and elemental equations are established. Finally in Chapter 9, MATLAB IFEM codes are developed for elements mentioned in Chapter 8, which are also illustrated through simple example problems. three-dimensional the ® As regards, good FEM books as well as few books on interval computing are certainly available. It may also be noted that interval finite element method is becoming a challenging tool to handle the uncertain problems. In this respect, various researchers throughout the globe have published good and interesting papers. While working on this field, we thought of having a book on this challenging area, which may help the students and researchers for their academic and industrial endeavor. Accordingly, the intended audience for this book must be the graduate, postgraduate, and doctoral students along with teachers, engineers, and researchers who are in need of handling the uncertain (interval) environment on different engi- neering subjects like civil, mechanical, aerospace and in science areas such as mathematics, applied and industrial mathematics, and physics. This book will serve to understand the interval uncertainties caused due to the vague or impreciseness, finite element method, and corresponding ® MATLAB codes. interval
Preface xi It is worth mentioning that very simple structural problems have been considered here to have the first-hand knowledge about handling interval ® uncertainties using IFEM. Interested readers may find the MATLAB codes very useful and fruitful because those are written in a very simple, systematic, and easy-to-understand form. The authors do believe that this book will cer- tainly ignite the users to write their own codes that may handle the related problems. Sukanta Nayak Snehashish Chakraverty
ACKNOWLEDGMENTS This book is the outcome of our last 7 years’ rigorous study and research in the area of uncertainty modeling. The uncertainty issues are addressed here ® through interval theory and then the interval finite element with MATLAB codes is developed for first-hand use. We have been inspired from the work of Dr. Peter Kattan and acknowledge him with high regard. The authors are very much grateful to National Institute of Technology Rourkela, India, and University of Johannesburg, South Africa, for giving a platform to ini- tiate this work. The first author expresses his gratitude to his father Mr. Hadibandhu Nayak and mother Mrs. Shrimati Nayak who supported him in each step of his life and have been a constant source of inspiration. Also, he would like to thank his friend Peehu who encouraged him all the time. Finally, he is very much grateful to those who have been with him over the course of the years and both directly and indirectly helped him for the completion of this book. The second author would like to thank first his beloved parents. Next he would like to thank his wife Mrs. Shewli Chakraborty and daughters Shreyati and Susprihaa for their continuous love, support, and source of inspiration at all the time during the preparation of this book. We would like to express our sincere gratitude to the readers who will go through this book and enjoy the beauty of interval finite element method ® with MATLAB for various structural problems. The contributors and authors referred in this book are greatly appreciated. Finally, we would like to thank the Academic Press for enabling us to publish this book and to the team of the publisher who directly or indirectly provided us help and sup- port throughout this project. xiii
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